Totem Pole Pfc Matlab

Totem-2022-001

Proton reconstruction with the CMS-TOTEM Precision

Abstract

The Precision Proton Spectrometer (PPS) of the CMS and TOTEM experiments col- lected 107.7 fb−1 in proton-proton (pp) collisions at the LHC at 13 TeV (Run 2). This paper describes the key features of the PPS alignment and optics calibrations, the pro- ton reconstruction procedure, as well as the detector efficiency and the performance of the PPS simulation. The reconstruction and simulation are validated using a sample of (semi)exclusive dilepton events. The performance of PPS has proven the feasibil- ity of continuously operating a near-beam proton spectrometer at a high luminosity hadron collider.

© 2023 CERN for the benefit of the CMS and TOTEM Collaborations. CC-BY-4.0 license *See appendices A and B for lists of collaboration members.

1

Introduction . .

The Cms Detector And Pps

.

3

LHC optics and proton transport . .

4

4

Data sets . .

5

5

Alignment . .

Alignment Fill

.

5.2

Physics fills . .

5.3

Timing RPs . .

6

Optics model and calibration . .

Introduction

.

Calibration Of The Lhc Optics

.

6.3

Optics description and uncertainty model . .

8

Aperture constraints . .

9

Proton simulation . .

Uncertainties

.

Validation With Dimuon Sample

.

12

PPS tracking efficiency . .

12.1

Silicon strip detector efficiency . .

12.2

Pixel detector efficiency . .

12.3

Multi-RP efficiency factor . .

13

Timing . .

Summary

.

The Cms Collaboration

.

Introduction

The Precision Proton Spectrometer (PPS) detector system has been installed and integrated into the CMS experiment during Run 2 of the LHC with 13 TeV proton-proton collisions. It is a joint project of the CMS and TOTEM Collaborations and measures protons scattered at very small angles at high instantaneous luminosity . The scattered protons that remain inside the beam pipe, displaced from the central beam orbit, can be measured by detectors placed inside movable beam pipe insertions, called Roman pots (RP), which approach the beam within a few mm. The PPS detectors have collected data corresponding to an integrated luminosity of 107.7 fb−1 during the LHC Run 2, which occurred between 2016 and 2018.

The physics motivation behind PPS is the study of central exclusive production (CEP), i.e. the process pp →p(∗) + X + p(∗) mediated by color-singlet exchanges (e.g. photons, Pomerons, Z bosons), by detecting at least one of the outgoing protons. In CEP, one or both protons may dissociate into a low-mass state (p∗); dissociated protons do not produce a signal in PPS. The X system is produced at central rapidities, and its kinematics can be fully reconstructed from the 4-momenta of the protons, thereby giving access to standard model (SM), or beyond SM (BSM) final states that are otherwise difficult to observe in the CMS central detectors because of the large pileup (multiple interactions per bunch crossing) at high luminosities. CEP provides unique sensitivity to SM processes in events with Pomeron and/or photon exchange, and BSM physics, e.g. via searches for anomalous quartic gauge couplings, axion-like particles, and new resonances .

This paper is organized as follows. The CMS detector and PPS are described in Section 2. The LHC optics and the concept of proton transport is presented in Section 3, followed in Section 4 by a description of the data sets used. Sections 5 and 6 describe the detector alignment proce- dure and the LHC optics calibration. Section 7 details the proton reconstruction with the PPS detectors. Sections 8 and 9 document the study of LHC aperture limitations and the simulation of the proton transport and PPS detectors, and Section 10 describes the uncertainties affecting the proton reconstruction. A validation of the reconstruction using a (semi)exclusive dimuon sample is presented in Section 11. The measurement of the proton reconstruction efficiency is discussed in Section 12. Section 13 describes a study of the performance of the proton ver- tex matching criteria from time-of-arrival measurements. Finally, a summary is presented in Section 14.

The Cms Detector And Pps

The central feature of the CMS apparatus is a superconducting solenoid of 6 m internal diam- eter, providing a magnetic field of 3.8 T. Within the solenoid volume are a silicon pixel and strip tracker, a lead tungstate crystal electromagnetic calorimeter, and a brass and scintillator hadron calorimeter, each composed of a barrel and two endcap sections. Forward calorime- ters extend the pseudorapidity coverage provided by the barrel and endcap detectors. Muons are measured in gas-ionization detectors embedded in the steel flux-return yoke outside the solenoid.

Events of interest are selected using a two-tiered trigger system. The first level (L1), composed of custom hardware processors, uses information from the calorimeters and muon detectors to select events at a rate of around 100 kHz within a fixed latency of about 4 µs . The second level, known as the high-level trigger (HLT), consists of a farm of processors running a version of the full event reconstruction software optimized for fast processing, and reduces the event rate to around 1 kHz before data storage .

3

Figure 1: Schematic layout of the beam line between the interaction point and the RP locations in LHC sector 56, corresponding to the negative z direction in the CMS coordinate system and the outgoing proton in the clockwise beam direction. The accelerator magnets are indicated in grey and the collimator system elements in green. The horizontal RPs, which constitute PPS, are marked in red. The vertical RPs are indicated in dark grey; they are part of the TOTEM ex- periment. The vertical RPs are not used during high luminosity data taking; nevertheless, they provide PPS with a reference measurement for the calibration and alignment of the detectors.

A more detailed description of the CMS detector, together with a definition of the coordinate system used and the relevant kinematic variables, is reported in Ref. .

The Pps Detectors

Figure 1 shows the layout of the RP system installed at around 200–220 m from the CMS inter- action point (LHC interaction point 5 (IP5)), along the beam line in the LHC sector between the interaction points 5 and 6, referred to as sector 56. A symmetric set of detectors is installed in LHC sector 45. Some RPs approach the beam vertically from the top and bottom, some hori- zontally. During standard machine operation, scattered protons undergo a large displacement in the horizontal direction and a small vertical displacement at the RP positions. The horizontal RPs are hence used. The vertical RPs are used in special configurations of the machine and in low luminosity proton-proton fills for the calibration and alignment of the detectors.

Each detector arm consists of two RPs instrumented with silicon tracking detectors that measure the transverse displacement of the protons with respect to the beam, and one RP with timing detectors to measure their time-of-flight. The tracking RP closer to the IP5 is referred to as “near”, the other as “far”. Silicon strip sensors with a reduced insensitive region on the edge facing the beam were initially used . Each RP housed 10 silicon strip sensor planes, half at a +45◦angle and half at a −45◦angle with respect to the bottom of the RP. These sensors could not sustain a large radiation dose and could not identify multiple tracks in the same event.

For this reason they have been gradually replaced by new 3D silicon pixel sensors: one RP (in each arm) during the 2017 data-taking run and all tracking RPs in 2018 were instrumented with 3D pixel sensors. Each such RP hosts six 3D pixel sensor planes . A summary of the RP configurations used in 2016-2018 is provided in Table 1.

The difference between the proton arrival times in the detectors on both sides of the IP5 is used to reject background events with protons from pileup interactions, or beam-halo particles. Timing detectors were operational in 2017 and 2018, with four detector planes hosted in a single RP. They consisted of single- and double-sided single crystal chemical vapor deposition (scCVD) diamond sensor planes ; during 2017 data taking one of the four planes consisted of ultra-fast silicon sensors instead of diamond ones.

4

Table 1: RP configurations in different years. The numbers represent the RP distances from the IP5, the sensor technology is indicated in parentheses. The RP layout was always symmetric about the IP5. There were always two tracking RPs per arm; the one closer to the IP5 is denoted as “near”, the other as “far”. In 2016, no timing RPs were used.

Lhc Optics And Proton Transport

PPS is a proton spectrometer that uses the LHC accelerator magnets between the interaction point (IP) and the RPs. Scattered protons are detected in the RPs after having traversed a segment of the LHC lattice containing 29 main and corrector magnets .

Since the protons that reach the PPS detectors travel more than 200 m inside the vacuum pipe of the LHC and very close to the LHC beams, we use the technique normally employed to model beams inside an accelerator. The trajectory of the protons in the vicinity of the central orbit [15, 16] can be described as follows. The proton kinematics d at a distance l from the IP (e.g. at the RPs) is related to the proton kinematics at the IP, d∗, via the transport equation: d(l) = T(l, ξ) · d∗.

(1)

Superscript ∗in general is used in the following to denote the value of the given parameter at the interaction point, z = 0. The proton kinematics is described by d = (x, θx, y, θy, ξ)T, where (x, y) and (θx, θy) indicate the transverse position and angles; ξ denotes the fractional

(2)

where pnom and p are the nominal beam momentum and the scattered proton momentum, respectively [17, 18]. In exclusive reactions the momentum losses of the two scattered protons, ξ1 and ξ2, can be used to assess the mass of the centrally produced state

(4)

where Ecm stands for the proton-proton centre-of-mass energy (13 TeV in LHC Run 2).

(5)

where the most important quantity for the proton spectrometer is Dx, the horizontal disper- sion; the other matrix elements are the so-called optical functions (vx, Lx, mi,j and their vertical

0.08

Figure 2: Frequency distributions of β∗vs. crossing angle configurations as extracted from data. Left: year 2017. Right: year 2018. counterparts) . The definition of the relevant optical functions and their determination are described in Section 6. The optical functions depend on LHC parameters like the betatron func- tion value β∗at the IP5 and the crossing angle. Throughout this document, we refer to the half crossing angle, i.e. half the angle between the beams at their crossing point.

Figure 2 shows the distributions of β∗vs. crossing angle for different data taking periods as extracted from data certified for analysis. In 2017, most of the data were recorded at four discrete values of the crossing angle: 150, 140, 130 and 120 µrad. The highest value was used at the beginning of the fills, then the crossing angle was reduced as the instantaneous luminosity dropped. The value of β∗was set to 0.4 m (0.3 m) in periods before (after) Technical Stop 2 (TS2). In 2018, the crossing angle was changed continuously from 160 µrad at the beginning of the fill down to 130 µrad. At this point, β∗was changed in two discrete steps, from 0.3 to 0.27 and finally to 0.25 m. In 2016 (not shown in the figure) β∗= 0.4 m was used together with the crossing angle values of 185 µrad and 140 µrad for the pre-TS2 and post-TS2 periods, respectively.

Data Sets

Two types of data are used for the calibration and alignment of the PPS detectors: data taken in high-intensity LHC “physics” fills and data taken in special “alignment” fills. The low beam intensity is an essential feature of the alignment fills, which provide additional data for align- ment and optics calibration. The various beam intensities are typically achieved by injecting various numbers of bunches in the LHC, since the number of protons per bunch is typically the same, up to 1.2 × 1011. The RP distances from the LHC beams are typically expressed in multiples of “beam sigmas”, the RMS values of the beam transverse profile. The values of the beam sigma are the same for the alignment and physics fills: σbeam ≈0.1 mm horizontally and σbeam ≈0.4 mm vertically.

The physics fills are standard LHC fills. There are up to 2500 bunches per beam, yielding an instantaneous luminosity of about 1034 cm−2 s−1. The average number of inelastic proton interactions at the IP (pileup) is typically between 15 and 55. Only horizontal RPs are inserted in these fills, to a distance of 15 σbeam.

“Far” Rp Unit

Figure 3: Illustration of a proton crossing both the vertical (blue) and the horizontal (green) RPs (overlapping configuration). The alignment fills use the same LHC optics as the physics fills, but much lower beam intensity— typically only two bunches are injected per each beam. This gives instantaneous luminosities of the order of 1030 cm−2 s−1 and average pileup about 20. The primary purpose of these fills is to establish the RP position with respect to the LHC collimators using a procedure analogous to the LHC collimator alignment . This is a precondition for systematic RP insertion close to the high-intensity LHC beams. Because of the low intensity, the safety rules allow insertion of both horizontal and vertical RPs very close to the beam: at 6.5 σbeam horizontally and at 5 σbeam vertically. At these distances, the horizontal and vertical detectors overlap, as shown in Fig. 3, which allows the relative alignment of the RPs in each arm. With the use of the vertical RPs, it is possible to detect elastically scattered protons that are used for horizontal RP alignment with respect to the beam. The alignment procedure is detailed in Section 5. In the alignment fills the very small separation of the horizontal RPs from the beam allows the recording of additional data essential for optics calibration (cf. Section 6). Typically there are two alignment fills per year of LHC operation.

In Run 2, PPS was operated from 2016 to 2018. The PPS data sets are divided in data-taking pe- riods. The PPS performance is often sensitive to the LHC settings (optics, collimators, etc.), which often vary with time; they are changed during LHC technical stops (TSs).

For In-

stance, the LHC optics was modified during the second technical stop (TS2) in 2016 and β∗ was changed after TS2 in 2017. The technical stops are also opportunities for changing the position of the detectors in the RPs. For example, in TS1 and TS2 in 2018, the tracking RPs were shifted vertically to better distribute the radiation dose accumulated by the pixel sensors.

The sensor inefficiency due to radiation damage is discussed in Section 12. Table 2 summarizes the PPS periods with significantly different LHC/RP settings and the corresponding integrated luminosities .

Alignment

The alignment of the RPs is a multi-level procedure including aligning the sensor planes within each RP as well as aligning the RPs with respect to the LHC beam. This is one of the inputs for the proton reconstruction (discussed in detail in Section 7).

Although conceptually similar, the alignment of RPs is different from that of other CMS sub- detectors, because the RPs are moveable devices. At the beginning of each LHC fill they are stored in a safe position away from the beam. Only when the LHC reaches stable conditions are they moved close to the beam. Since the fill-to-fill beam position reproducibility has a limited accuracy, it is desirable to determine the alignment parameters for every fill.

7

Table 2: List of the PPS periods with distinct LHC and/or RP settings. The third column from left indicates the time ranges where PPS recorded data. Lint corresponds to the integrated luminosity recorded during runs certified for use in physics analysis.

Pre-Ts2

4974 (31 May) to 5052 (29 Jun), 5261 (29 Aug) to 5288 (9 Sep)

107.7

The alignment procedure involves multiple steps. A special “alignment” calibration fill deter- mines the absolute position of the RPs with respect to the beam (Section 5.1). This calibration then serves as a reference for the alignment of every “physics” fill with standard conditions (Section 5.2). Once the tracking RPs are aligned with respect to the beam, the timing RPs are aligned with respect to the tracking RPs (Section 5.3).

Alignment Fill

An alignment fill is a special fill, which allows to obtain data essential for calibration, not avail- able in standard physics fills (more details are given in Section 4). The relative alignment among the sensor planes in all the RPs and among all the RPs in one arm is determined by minimizing residuals between hits and fitted tracks . This is an it- erative procedure, since a priori it is not possible to distinguish between misalignments and outliers (unrelated hits due to noise, etc.). Therefore, the iteration starts with a large toler- ance, O(100 µm), that allows for misalignments, and as it proceeds the tolerance is decreased to O(10 µm) as outliers are discarded. An illustration is shown in Fig. 4, left, emphasizing the essential role of the overlap of the vertical and horizontal RPs. The typical uncertainty of the relative RP alignment is few micrometres. By construction, the relative alignment is not sen- sitive to misalignment modes that do not generate residuals, e.g. a global shift of the full RP system. These modes are addressed in the next step.

The vertical RPs can detect protons from elastic scattering, i.e. a process with only two protons in the final state, each having ξ ≡0 as a consequence of momentum conservation. Because the two protons emerge from the same vertex in opposite directions, elastic events are relatively easy to tag (cf. Section 5.2.1 in Ref. ). Because of the azimuthal symmetry of the elastic scat- tering at the IP and the properties of the LHC optics, the elastic protons arrive at the RPs with impact points in the transverse plane elliptically distributed around the beam. Although only the tails of the elastic hit distributions are within the acceptance (protons with sufficiently large

Vertical Scattering Angle, |Θ∗

y|), the distributions can be used to extract the beam position with respect to the RPs. This is illustrated in Fig. 4, right: the profile of the elastic hit distribution (black) is interpolated between the top and bottom RP (green), which provides information on the horizontal alignment and potential rotations in the xy plane. This is combined with the in- formation from a minimum bias sample, in which most protons detected in the horizontal RPs

2016 (13 Tev)

Figure 4: Left: relative alignment between vertical and horizontal RPs (April 2018). The plot shows track impact points in a scoring plane perpendicular to the beam. The points in red represent tracks only reconstructed from vertical RPs, in blue only from horizontal RPs and in green from both vertical and horizontal RPs. The size and position of the RP sensors is schematically indicated by the black (vertical strip RPs) and magenta (horizontal pixel RPs) contours. Right: determination of the beam position with respect to the RPs (September 2016).

Black: profile (mean x as a function of y) of elastic track impact points observed in vertical RPs; green: fit and interpolation. Blue: horizontal profile of minimum bias tracks found in the horizontal RP; red: fit and extrapolation. Magenta cross: the determined beam position. The error bars represent statistical uncertainties.

are due to pileup. The profile from the minimum bias sample (blue) is extrapolated linearly (red) to find the intersection (magenta cross) with the green line. The intersection indicates the beam position with respect to the RPs, with a typical uncertainty of about 10 µm.

Physics Fills

For each high-luminosity LHC fill (“physics” fill), the horizontal RP alignment is obtained by matching observations from the fill to those from the reference “alignment” fill, cf. Section 5.1. Various matching metrics have been used, and some of the first choices are discussed

(6)

where yN and yF stand for the vertical track positions in the near and far RP, respectively. Similarly, ytest refers to the vertical track position in the RP being aligned. The shape of the profile is illustrated in Fig. 5, where the value of S corresponds to the slope of the red line.

The x dependence of the S function is generated by the LHC optics, cf. Section 6: y is mostly given by the vertical effective length, Ly(ξ), and ξ is largely correlated with x because of the large horizontal dispersion. The optics has been verified to be stable in time and therefore S(x) is suitable for matching observations between different fills. Furthermore, the function from Eq. (6) is convenient because of its slope character: vertical misalignments (shifts in y) cause no bias and unavailable parts of the phase space (e.g. because of localized radiation damage) do not have any detrimental impact since the slope can still be determined from the available part. The matching procedure is illustrated in Fig. 6, left: the S(x) curve from the test fill (blue) is shifted left and right until the best match with the S(x) curve from the reference fill (aligned

2018 (13 Tev)

Figure 5: Illustration of yF −yN dependence on y (fill 7139, 2018, near RP in sector 56). The three plots correspond to three x selections as indicated in the legends. Blue: profile histogram of the dependence, red: linear fit to the central part. The error bars represent statistical uncertainties.

with the method from Section 5.1) is found. The shift between the blue and red curves is then used as the alignment correction. The relative alignment between the RPs within the same arm is then refined with a dedicated method with a better sensitivity — good calibration of the relative alignment is essential for some of the proton reconstruction techniques. The relative near-far alignment method is based on comparing horizontal track positions in the near and far RPs, xN and xF, respectively. The procedure is illustrated in Fig. 6, right: the profile xF −xN vs. xN (red) is extrapolated (blue dashed) to the value of xN corresponding to the beam position (green). The extrapolated value of xF −xN (magenta dot) then gives the relative-alignment correction. In general, the xF −xN difference can be generated either by misalignments (independent of the horizontal position) or by the optics (roughly proportional to horizontal displacement from the beam). The extrap- olation to the beam position, where the displacement from beam is ≈0, thus suppresses the optics contribution and keeps the misalignment component only.

The vertical alignment is obtained by extrapolating (blue) the observed vertical profile (red) to the horizontal beam position (green), as shown in Fig. 7 where the alignment correction is marked with the magenta dot. The extrapolation to the beam position suppresses the optics contributions and keeps the misalignment component only. The mode (most frequent value) of y, contrary to the mean of y, is a local estimator not considering the tails of the y distribution, which can be truncated because of the limited sensor size or other acceptance related effects.

This vertical alignment method is sufficiently sensitive to provide both absolute per-RP and relative near-far alignment. Figure 8 shows a summary of per-fill alignment results for one alignment period. It also illus- trates one of the many systematic validations performed; compatible results are expected from data sets obtained with different values of the crossing angle, β∗, or different central-detector triggers (the vast majority of the protons reaching the RPs are due to pileup unrelated to the triggering event).

Figure 8 also confirms the expectation of fill independence of the alignment results. A fit of the results is used to remove occasional outliers, improve fill-to-fill stability and increase the overall accuracy of the alignment. In Run 2, there were two alignment periods where significant time variation was observed for some RPs. A notable example is 2016 pre-TS2 (additional details are discussed in Section 3.5 of Ref. ) where a package of sensors was initially wrongly inserted

2018 (13 Tev)

Figure 6: Left: illustration of the absolute horizontal alignment (fill 5424, 2016 post-TS2, far RP in sector 45). Black: data from the reference alignment fill, blue: data from a physics fill before the alignment and red: data from the physics fill, aligned to match with the black reference.

The error bars represent the bin sizes (horizontally) and statistical uncertainties (vertically). Right: illustration of horizontal near-far relative alignment (fill 7052, 2018 and sector 45). Red: mean value of xF −xN as function of xN. Blue: fit and extrapolation to the horizontal beam position (vertical green line, e.g. from the left plot). The value of the relative near-far alignment correction is indicated by the magenta dot.

2016 (13 Tev)

Figure 7: Illustration of the vertical alignment (fill 5424, 2016 post-TS2, far RP in sector 45). Red: mode (most frequent value) of y as a function of x, Blue: fit and extrapolation to the horizontal beam position (indicated by the vertical green line and extracted from Fig. 6, left). The value of the vertical alignment correction is indicated by the magenta dot. The error bars represent the systematic uncertainties.

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into a RP and over time the package slowly drifted to its nominal position due to the spring included in the RP assembly. Even in these cases, the variation was slow enough that fits could be applied to suppress the excessive fill-to-fill fluctuations and thus improve the results.

The alignment uncertainties are presented in Table 3. They are estimated from fill-to-fill result fluctuations in cases where identical results are expected. Table 3: Summary of per-fill alignment uncertainties.

Timing Rps

The timing RPs consist of four sensor layers, called “planes”, perpendicular to the LHC beam. As shown in Fig. 9, each plane is composed of four physical pieces of diamond substrate, called “chips”. Each chip has a structure of readout electrodes in the form of thick vertical strips, called “pads”. This structure constitutes the horizontal segmentation of the timing detector and, in general, is different for each plane and chip.

The timing sensors are aligned with respect to the tracking RPs to associate local tracks using timing and tracking RPs (cf. Fig. 28). Since the timing RPs have only horizontal segmentation, only x alignment is performed. The alignment is performed individually for each plane and pad as well as for each LHC fill.

As illustrated in Fig. 10, the alignment method is based on a histogram of horizontal residuals between the hit position in the timing sensor and the track interpolated from the upstream and downstream tracking RPs. The histogram of these residuals (red) reveals the “shape” of the pad, the pad edges (dashed blue) as well as the pad centre (green). The alignment correction is given by the offset of the green line from zero. Estimated correction uncertainty is 100 µm, driven by the uncertainties of the extracted pad edge positions.

A typical example of alignment corrections is shown in Fig. 11. As expected, we find compat- ible results for the pads on the same physical chip, cf. Fig. 9. The average per-chip correction is indicated by the short horizontal line. The result pattern can be explained by the mechan- ical process of gluing the chips on the board — the chips cannot mechanically overlap, only additional gaps can be introduced. This leads to a cumulative misalignment monotonically in- creasing (in absolute value) with the chip number, as revealed by the results. Chip 3, the most far from the beam, often gets an insufficient number of tracks (because of the LHC collimators, cf. Section 8) and the correction from chip 2 is used in this case.

2017 (13 Tev)

Figure 8: Example of per-fill alignment results (2017 post-TS2, sector 56 or near RP in sector 56). Horizontal axis contains representative LHC fills where PPS was active. The colors in- dicate two values of crossing angle, 120 µrad (blue) and 150 µrad (green). Rows from top to bottom: absolute horizontal alignment, near-far relative horizontal alignment, absolute vertical alignment and near-far relative vertical alignment. The error bars (mostly invisible) represent a combination of statistical and systematic uncertainties.

Chip 3

Figure 9: Example of timing detector segmentation in one plane (plane 1 in 2018 configuration). The beam is at x = 0 mm. Chip boundaries are drawn as dashed black rectangles. Pads are visualized as thick vertical strips, their colors indicate the chip relation.

2018 (13 Tev)

Figure 10: Illustration of the timing-RP alignment method (fill 7137, 2018, sector 56, plane 1 and pad 9). The red histogram shows the difference between the horizontal track position in the timing sensor, xtiming, and the track interpolated from the tracking RPs, xtracker. The vertical blue dashed lines indicate the identified pad boundaries, the green line the pad center.

Chip 3

Figure 11: An example of alignment corrections in a single timing RP sensor plane (fill 7137, 2018, plane 1). Two different markers are used: the dots represent per-channel measurements, while the short horizontal lines represent per-chip averages. The same color is used for chan- nels/pads placed on the same diamond chip, following the scheme in Fig. 9. For chip 3 (most far from the beam) sometimes the track statistics is insufficient for alignment determination.

In such cases the magenta thick dot is missing. The error bars represent a combination of sta- tistical and systematic uncertainties reflecting the sharpness of the pad boundaries shown in Fig. 10. Left: sector 45, right: sector 56.

Introduction

In Run 2, the LHC optics settings and conditions were modified every year. The key concepts and the tools to constrain the main optical functions using collision data for 2016 have been described in Refs. [17, 18]. During physics runs, the luminosity of the LHC beams decreases naturally due to bunch intensity decay. Luminosity can be regained for the experiments by adjusting the crossing angle and betatron amplitude to increase the so-called luminosity geom- etry factor. To achieve this goal in 2017 the levelling of the crossing angle and of the betatron amplitude β∗was introduced. In 2018 the levelling of both parameters became continuous .

The modelling of this varying optics and its calibration required a generalization of the well- established 2016 methods; the higher number of events permitted, and also required, a more careful dispersion calibration. The vertical position of the beams crossing point, y∗, also changed with respect to 2016. In the last two years of Run 2, the optics had a sizable vertical dispersion Dy, which is an important optical function for the reconstruction. An optics uncertainty model based on collision data is also presented. The optics calibration methods of Run 2 are briefly discussed from the viewpoint of the HL-LHC in Ref. .

Proton Transport At The Lhc

The transport matrix and the optical functions have already been introduced in Section 3. In the following, the meaning of the transport matrix elements is explained, with emphasis on the connection between the β amplitude and the optical functions used in the reconstruction.

Specifically, the horizontal and vertical magnifications

(8)

are functions of the betatron amplitudes βx,y, their value β∗at IP5 and the relative phase ad-

Βx,Y

.

(9)

The beam size can be calculated from the beam emittance ε of the LHC and from the betatron

(10)

using a representative βx = 0.3 m value, where ε is computed from the normalised emittance εN = (βL γL)ε = 3.75 µm rad. Here βL = v/c; v is the velocity of the beam particles, c is the

L)−1

2 is the Lorentz factor. The subscript “L” is used in βL and γL to avoid confusion. The Liouville theorem dictates that

(11)

where σ(x′) is the beam divergence, i.e. the angular spreading, of the LHC beams; the symbol x′ stands for dx/dl . Therefore, from Eq. (11) it follows that σ(x′) =

X Ε ≈40 Μrad For

the representative βx = 0.3 m value, which gives the limit on the resolution of the scattering

Angle Θ∗

x,y of PPS .

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As already mentioned, in 2017, the necessity to improve the lifetime of the beams led to the change or “levelling” of both the betatron amplitude, β∗, at IP5 in discrete steps and the hori- zontal crossing angle. In 2018 both parameters were modified continuously (cf. Table 4). For comparison at IP1 (ATLAS) the crossing angle bump was in the vertical plane during Run 2 to avoid long range beam-beam interactions . The levelling is based on the so-called Achro- matic Telescopic Squeezing (ATS) optics ; one of its features is that the optical functions Eq. (7) and Eq. (8) remain constant despite the change in β∗. Therefore, the β∗levelling is a transparent operation from the viewpoint of the reconstruction. The horizontal dispersion Dx determines the proton trajectory in the horizontal plane and depends on the crossing angle levelling at IP5; therefore Dx is calibrated separately for each reference crossing angle.

Table 4: Summary of main beam parameter values, crossing angle and β∗, during the Run 2 period per year. In 2017 the values changed in discrete steps, whereas in 2018 there was a continuous change within the interval.

[0.25, 0.4]

The transport equation Eq. (1) can be explicitly written at the RPs in the form

(12)

that describes the connection between the proton kinematics at the IP5 and at the RPs, where x0 and y0 are the horizontal and vertical beam position, respectively. The horizontal dispersion Dx is a function of ξ, therefore it is useful to define a function that provides the horizontal position

(13)

and one can define similarly yd(ξ) [2, 29]. The coupling terms mij in the transport matrix Eq. (5) connect the horizontal and vertical scat- tering planes. At the LHC, like for most accelerators, these terms are set to zero nominally m13, . . , m42 ≈0 for collision optics. They receive perturbative-level corrections because of skew quadrupole corrector magnets. The effect of the coupling on the reconstruction of the proton kinematics was negligible for all years.

The optics calibration assumes the beam-based alignment of the detectors, after which the beams appear at x0 = y0 = 0, cf. Eq. (12) . The horizontal position of the protons, x(ξ), is a nonlinear function of ξ, which can be approximated for low ξ values

(14)

where the resolution in x is limited by the spreading because of the scattering angle term Lx · θ∗

X

and by the contribution of the vertex x∗, cf. Eq. (12).

Calibration Of The Lhc Optics

The horizontal dispersion Dx is the most important optics quantity, because it allows one to convert the x-coordinate measurements at the RPs into the fractional proton momentum loss

17

ξ. The determination of Dx from the measured proton tracks is briefly reviewed in the next section (cf. also Ref. ). The 2017 and 2018 optics calibration procedure goes a step further and also exploits (semi)-exclusive µµ production; the exclusivity of the process plays a key role in the calibration, as illustrated in Section 11.

In the last step of the calibration procedure, the vertical dispersion Dy is determined from min- imum bias RP data. The calibration of the dispersion functions is followed by the calibration of the remaining optical functions in the transport matrix Eq. (5), namely the horizontal, Lx(ξ), and vertical, Ly(ξ), effective lengths, and the corresponding magnification functions; other op- tical functions are less relevant for the proton reconstruction.

The above optics calibration steps rely on the nominal transport model, which is taken from LHC databases. The transport matrix is defined by the machine settings M, which are obtained from several data sources. The proper version of the LHC magnet lattice description, known as “sequence”, is used each year. The nominal magnet strength file for a given beam optics is always updated using measured data: the currents of the magnets power converters IPC are first retrieved using TIMBER , an application to extract data from heterogeneous databases containing information about the whole LHC infrastructure. The currents IPC are converted to magnet strengths with the LHC software architecture (LSA) , which uses the conversion curves from the field description for the LHC (FIDEL) .

The Ly = 0 Method

This procedure uses the minimum bias data recorded during the special low-luminosity runs mentioned in Section 5.1. The method has been applied for each year within Run 2; for 2017 and 2018, a separate calibration was carried out for each crossing angle. The procedure assumes the calibration of the vertical effective length Ly for low-ξ values, below ξ ≈4%, using the elastic candidate events measured in the vertical RPs; this additional step is reported in detail in Refs. [17, 33].

The LHC optics are calculated with the methodical accelerator design (MAD-X) program, a general purpose beam optics and lattice software . The vertical effective length Ly(ξ) is a function of the proton momentum loss ξ, and can be calculated with MAD-X at each RP location with good accuracy. The calibration is based on the observation that Ly(ξ) is positive at ξ = 0, monotonically decreases with increasing ξ reaching large negative Ly values and it vanishes at about ξ ≈4%. According to Eq. (12) at this ξF value every proton is transported to the same vertical coordinate y = 0 regardless of the vertical scattering angle θ∗

Y (The Vertex

contribution is neglected). At the same time these protons appear at the horizontal location xF ≈Dx · ξF. Consequently, the (x, y) distribution of the protons has to exhibit a “pinch”, or focal point, at this horizontal location xF, cf. Fig. 12.

The LHC optics transport is the same for all protons, thus the focal point can be observed and measured with the horizontal RP detectors using large statistics minimum bias data, cf. Fig. 13. The figure shows the (x, y) distribution of the proton impact points in the RP detectors for 2017 for a representative half crossing angle αh = 120 µrad. The plot shows the parabolic fit of the contour curves around the “pinch” point. The minima of the parabolic curves are fitted with a linear function and the fits are extrapolated. The intersection of the linear fits is marked with a red dot, and indicates the estimate of the focal point position xF. The fit of the contour lines and the extrapolation are used in order to estimate the bias coming from the scattering angle θx and extrapolate to the point where the bias vanishes. The measurement is repeated with the distribution obtained after a selection on the scattering angle θx to reduce the horizontal spreading around the focal point; in this case the parabolic fits are not needed.

Tracks

Figure 12: The (x, y) distribution of simulated proton tracks in the near RP in sector 56 using MAD-X. It illustrates the “pinch” or focal point at x = xF where the vertical effective length vanishes: Ly(ξF) = 0, given the relation y ≈Ly(ξF) · θ∗

Y. The Simulation Takes Into Account That

the small vertical dispersion moves particles upward according to ∆y = Dy ξ with increasing x, and ξ.

Ξf

.

(15)

The measured Dx values are used to calibrate the LHC optics model, as described in the next sections. The uncertainty of the Ly = 0 method includes the uncertainty of the contour fits, their minimum and their linear extrapolation; the systematic uncertainty due to remaining bias is estimated with a Monte Carlo simulation.

Calibration Using The (Semi)-Exclusive Μµ Process

In 2016 PPS collected its first (semi)-exclusive dilepton sample , pp →p(∗)ℓ+ ℓp(∗), where a pair of leptons (ℓ= e, µ) is reconstructed in the central CMS apparatus, one of the protons is detected in PPS, and the second proton either remains intact or is excited and then dissociates into a low-mass state, indicated by the symbol p(∗), and escapes undetected. Section 11 focuses on the µµ measurement, whereas the implications on the optics calibration are presented here.

The (semi)-exclusivity implies a high-purity data set: in these events, the central µµ system carries the momentum lost by the two forward protons. Therefore, the difference of the frac- tional momentum loss reconstructed from PPS and from the central CMS detectors can be de- termined; the correction to Dx is computed such that this difference vanishes. The improved calibration result for Dx remains within the uncertainty of the Ly = 0 method and the final Dx result is the weighted average of the two measurements. The uncertainty of Dx is the combined uncertainty of the Ly = 0 and the (semi)-exclusive µµ methods. The evolution of the dispersion Dx(ξ) (or xd(ξ) cf. Eq. (13)) with ξ can be also validated using the µµ results. The Dx results are shown in Table 5 with a conservative 8% uncertainty in Dx, which applies to xd as well.

The dispersion asymmetry between the two arms was observed in 2016 and persisted in 2017 and 2018 as well; it is attributed to crossing angle asymmetry and quadrupole magnet mis- alignment within their nominal tolerance.

Events

Figure 13: The (x, y) distribution of the proton impact points in the near RP detector in sector 45 for 2017 using minimum bias data, along with parabolic fits of the contours around the “pinch”. The minima of the parabolas are fitted with a straight line. The intersection of the two lines is marked with a red dot, and indicates the estimated focal point coordinate xF. To make the contour curves and extrapolation symmetric, the mean of the histogram was aligned to 0 to remove the y offset created by the vertical dispersion ∆y = Dy ξ. The vertical error bar on the contour minima, blue points, represents the statistical uncertainty of the fit.

Table 5: Measured horizontal dispersion values Dx in the near RP at low ξ between 2% and 4% (the exact ξF value depends on the detector and the year). The resulting Dx value is the weighted average of the Ly = 0 and (semi)-exclusive µµ results. The quoted 8% uncertainty in Dx applies to the xd function as well.

Optics Matching

The purpose of the optics fitting (or “matching”) is the calibration of the LHC optics model us- ing the measured dispersion values and other measured constraints. The calibration procedure consists of a χ2 minimization with MINUIT, where the initial optics model of the fit is taken from the LHC databases, as mentioned in Section 6.2 .

The first step is to constrain the quadrupole field model using the elastic candidates from the alignment fills, described in Ref. . In the second step the measured dispersion values from Table 5 are used as inputs to the χ2 function, with additional constraints reflecting the LHC

Design + Χ2

measured.

(16)

The following measurements from both LHC beams contribute to χ2

Measured:

• the readings of three beam position monitors (BPMs) (at l = 22 m, 58 m, 199 m), with

20

• the beam position at RP 210, near, vertical, with an uncertainty σx = 0.5 mm; • the two measured dispersion values Dx (1 per arm) with their measured uncertainty, cf. Table 5.

To match, or fit, the dispersion values and the LHC optics model, the relevant LHC machine parameters are varied during the minimization. The matching procedure exploits the fact that a quadrupole magnet misaligned by a δx offset gives a correction to the dipole field, whereas the quadrupole fields remain unchanged. The following machine parameters have to be matched for the two LHC beams separately to obtain the orbit model for the proton reconstruction:

• Horizontal (Half) Crossing Angle Αh;

• quadrupole positions (σx = 0.5 mm, 6 parameters); • kicker strength (σk ≈3%, 3 parameters). With this procedure a good confidence level was achieved for the lattice model of the two LHC beams. The matched MAD-X optics model is used to extend the measured dispersion values from Table 5 to higher ξ values. An example of the fitted result is shown in Fig. 14.1

2017 (13 Tev)

Figure 14: The momentum loss of the protons ξ as a function of x in the near RP of sector 45. The dispersion function is ξ dependent itself and the figure shows directly the nonlinear x(ξ) = D(ξ) · ξ function. The ξ(x) function depends on the crossing angle as well; the figure shows the dependence for three reference angles, so the function can be interpolated to arbi- trary intermediate angles.

The optics model MAD-X shows that the different interpretations of the dispersion asymme- try between sector 45 and 56 (crossing angle rotation, quadrupole misalignment, etc.) lead to negligible differences in the systematic uncertainty, for example in the evolution of Dx with ξ.

Calibration Of The Vertical Dispersion Dy

In 2016 the vertical dispersion Dy was close to zero, whereas in 2017 and 2018 the optics changed and a vertical dispersion Dy ≈−1 cm was applied. Despite its small value, the verti-

21

cal dispersion has a strong effect on the ξ dependence of the vertical reconstruction of θ∗

Y And

y∗because of the nonlinearity of the other optical functions. The vertical dispersion Dy is estimated from the (Dy/Dx) ratio measured on the (x, y) plane; the value is refined by perturbing it so as to match the measured θ∗ y and y∗values as well.

Figure 15: The Vertical Scattering Angle Θ∗

y as a function of ξ after calibration of the vertical dispersion Dy for sector 45 and fill 6923. The mean of the scattering angle distribution is con- sistent with 0. The distribution is also affected by the vertical acceptance limitations starting from about ξ ≈5% because of the vertical acceptance limits of the detector cf. Fig. 3.

The measured vertical dispersion values are summarized in Table 6. The values are small enough that the crossing angle dependence can be neglected. The vertical dispersion values Dy are validated with minimum bias data, cf. Section 7 and also Fig. 15.

Table 6: Final measured vertical dispersion values Dy in the near RP per year. The uncertainty is derived conservatively from the measured (Dy/Dx) ratio.

Optics Description And Uncertainty Model

The LHC optics model, calculated with MAD-X, can be described in several efficient ways for the event reconstruction and physics analysis . In the year 2016, the description of the proton transport used orthonormal polynomials to fit the (x, y) coordinates of the protons at the RPs as a function of their input kinematics .

Experience with the data and optics modelling showed that the parametrization, or factoriza- tion, of Eq. (12) is sufficient to describe the proton transport between IP5 and the RPs; therefore, since 2016 an expansion using only 1-dimensional ξ dependent optical functions is applied.

As discussed earlier, in 2017 the levelling of the beam crossing angle was introduced. This is straightforward to take into account using the optical function concept with an additional

2017 (13 Tev)

Figure 16: The horizontal and vertical effective lengths Lx and Ly transport the scattering angle

X And Θ∗

y to the position (x, y) at the RPs. The figure shows the ξ de- pendence of the two functions. The horizontal effective length Lx(ξ) decreases faster than the vertical function Ly; both of them cross zero at low ξ, below ξ = 5%. The grey dashed lines show the effective lengths for the TOTEM RPs used for calibration.

extrapolation function among reference crossing angles, as shown in Eq. (17) and Fig. 14:

X(Αh, Ξ) = X120(Ξ) + 120 −Αh

120 −140 [x140(ξ) −x120(ξ)] .

(17)

The linear function is motivated by MAD-X and is compatible with the dispersion measure- ments within uncertainties. The other optical functions remain constant during the levelling of the crossing angle and, due to the telescopic concept of the ATS optics, they also remain constant during the levelling of β∗. The relevance of the ATS telescopic squeezing from the viewpoint of uncertainty model is discussed in Section 6.3.1.

Optics Uncertainty Model

The uncertainties of the horizontal and vertical dispersions Dx and Dy, and of the function xd(ξ) have already been discussed in Sections 6.2 and 6.2.4 (cf. also Table 5). The uncertainties of the remaining relevant optical functions are illustrated in the following.

The levelling of the crossing angle and β∗, mentioned earlier, is based on the ATS optics, which has been conceived to cope with requirements expected for HL-LHC . The most important feature of the ATS optics, from the viewpoint of the forward spectrometers, is that the magnetic fields around the IP are kept stable during the levelling process. The β∗at these IPs is changed by varying the magnetic fields at IP2 and IP8 . This stability significantly reduces the un- certainty in the optics model and transport matrix for PPS. It also contributes to the alignment stability, which uses the distribution of Ly/[dLy/dl], cf. Eq. (6) and Eq. (12).

Despite its stability, the LHC is subject to additional imperfections ∆M, which alter the

Transport Matrix By ∆T:

T (l; M) →T (l; M + ∆M) = T (l; M) + ∆T.

23

The principles of the optics uncertainty model are described in Ref. . A more complex approach is however needed in view of the explicit ξ dependence of the optical functions. The transport of protons in the vicinity of the central orbit, or any other reference orbit with a certain ξ, is mainly determined by the quadrupole fields of the alternating focusing and de- focusing magnet (FODO) system of the LHC, whereas the position of the central orbit itself is determined by the distribution of the dipole fields; this includes the dipole fields created by misaligned quadrupole magnets.

2017 (13 Tev)

Figure 17: Fit of σ(∆x) as a function of x for the near RP in sector 45. The fit is used to estimate the uncertainty of the optical function dLx/dl. The vertical error bars represent the statistical uncertainties.

A typical example is the assessment of the uncertainty of the optical function dLx/dl. The estimation starts with the uncertainty model at low ξ; the magnet strengths in MAD-X are perturbed within their nominal uncertainty and the model is refined using the optics con- straints from elastic candidates. In the next step the ratio of the optical function is estimated between the low- and high-ξ part using collision data, cf. Fig. 17. The estimation is based on

The Relation ∆X(X1) = Dlx/Dl|X=X1 · Θ∗

x and exploits the fact that the scattering angle distri- bution of the proton is almost independent of ξ, so that R(x1, x2) = σ(∆x(x2))/σ(∆x(x1)) ≈ dLx/dl|x=x2/dLx/dl|x=x1 .

After careful evaluation for this particular function the optics model and the data agree within ≈10%, cf. Fig. 17. The R(x1, x2) result is translated to R(ξ1, ξ2) using the dispersion and, to- gether with the low-ξ uncertainty, determines the uncertainty at all ξ. A similar procedure leads to the uncertainty of Ly(ξ). The LHC optics give strict correlations between the magnifi- cations vx, vy and Lx, Ly. Therefore, the uncertainty estimation of the effective lengths indirectly provides uncertainties on the magnifications as well.

Covariances Of Optical Functions

To fully estimate the ξ dependence of the uncertainty of the optical functions, the calculation of the covariance matrix between different ξ values for each function is needed. The magnetic strength k and other relevant beam parameters are perturbed within their nominal uncertainty

24

and the optical functions are calculated for each parameter set. The values of the obtained optical functions Lx and the envelope function thus obtained are shown in Fig. 18. The covari- ance and correlation matrix for the optical function Lx at the fractional proton momentum loss ξ = 3% and ξ = 10% are shown in Table 7.

2018 (13 Tev)

Figure 18: Left: the distribution of the horizontal effective length Lx(ξ) values as a consequence of perturbations of the magnetic strength. Right: the correlations of the functions; the red and blue dashed curves represent the two extreme Lx(ξ)-curves of the Monte Carlo. The upper and lower envelopes demonstrate that the points of the curve move together at different ξ.

The correlation matrix, shown in Table 7, indicates a close to 100% correlation between the low- and high-ξ regions, which is included in the uncertainty model, cf. also Fig 18. This means that the variations of the magnetic strength and other beam parameters act in the same way at different ξ values and the uncertainty can be described with one parameter. The covariance and correlation matrices are available for all optical functions.

Table 7: The correlation matrix for Lx between different ξ values for the detector RP56-220-fr vertical.

2018 (13 Tev)

Figure 19: Left: ξ dependent uncertainty function of the horizontal effective length Lx(ξ). Right: ξ dependent uncertainty function of the derivative of the horizontal effective length dLx(ξ)/dl.

The optics uncertainty model includes the close to 100% correlation. This means that the optical function perturbation δo can be determined at a given reference ξref value and can then be scaled with the factor given in Fig. 19 to obtain the perturbation at a different ξ value. The optics uncertainty model is included in the PPS proton simulation described in Section 9.

Inversion Of The Proton Transport Equations

The transport equations Eq. (12) are linear in ξ and in the horizontal scattering angle θ∗

X With

coefficient functions like Lx(ξ), which are nonlinear. The beam size σ(x) from Eq. (10) mul- tiplied by the magnification factor |vx| ≈4 gives σ(x) vx ≈60 µm in the horizontal plane, a contribution that is negligible when compared with the other two terms. Therefore, Eq. (12)

(19)

where the optical functions, like Lx,near(ξ), are functions of ξ. The variable ξ appears on both sides of the first nonlinear equation, whose solution can be found with any iterative method. These formulae are equivalent to those developed and used previously by the TOTEM Collab- oration . Equation (19) indicates the optical functions whose calibration is most relevant for the reconstruction. The formulae for the vertical reconstruction read:

(20)

where y′ = y −Dy ξ. The nonlinear Eq. (20) shows that an otherwise constant offset in Dy, or in the vertical alignment would lead to a nonlinear distortion of the reconstructed angle.

Summary

In summary, the LHC optics settings and conditions changed every year in Run 2. In this chapter the main concepts and the data-driven tools to constrain the optical functions for 2016, 2017 and 2018 have been presented. The main challenges of Run 2 are the levelling of the instantaneous luminosity by changing the crossing angle and β∗, which requires the careful calibration of the horizontal dispersion Dx and also its change with the crossing angle. The vertical dispersion Dy became sizable in 2017 and 2018, and its calibration has been discussed.

An optics uncertainty model based on collision data has been also presented, which includes the covariance matrix of transport elements.

Proton Reconstruction

The proton reconstruction consists in back-propagating the protons from the RPs (where they are measured) to the IP (where the kinematics is determined). The propagation follows the LHC optics discussed in Section 6. The input to the propagation consists of the proton tracks detected by the RPs and aligned with respect to the LHC beam (cf. Section 5). Since the proton tracks at the RPs are linear (no magnetic field), they can be described by four independent parameters (slopes and intercepts along x and y). The five proton kinematic variables include: the transverse position of the proton at z = 0, x∗and y∗, the horizontal and vertical scattering

X And Θ∗

y, and the fractional momentum loss, ξ. Compared to the four parameters measurable by the RPs, the reconstruction problem is underconstrained and a variable must be fixed with external information. Two complementary reconstruction strategies are exploited: “single-RP” and “multi-RP”.

The single-RP reconstruction is a simple approach that uses information from single RPs only. Because of the reduced input information, only ξ and θ∗

(21)

where the value of ξ reconstructed from the former equation is inserted into the latter. These equations reflect only the leading terms from the optics decomposition in Eq. (12). Neglecting the subleading, but still relevant, terms (e.g. the one proportional to θ∗

X) Implies A Degraded

resolution. On the other hand, a notable advantage of this approach is its applicability even when the proton track is not available in the other RP of the arm. Furthermore, this approach has a different (slightly smaller) dependence on the systematic variations with respect to the multi-RP method, cf. Fig. 38. In this sense the single-RP reconstruction is a very useful check

X Cannot Be Reconstructed With This Approach And

they are set to zero. For the vertex coordinates this is a reasonable approximation when low β∗ optics is used (as detailed below). The multi-RP reconstruction exploits the full potential of the spectrometer: it searches for pro- ton kinematics that best match the observations from all RPs and all projections by minimizing

(22)

where i runs over all the tracking RPs in the arm and q over the two transverse projections. This expression follows the notation of Eq. (1): the vector di represents the (measured) proton position at the ith RP, the vector d∗denotes the proton kinematics at the IP and the matrix Ti stands for the proton transport between the IP and the ith RP. The quantity σi

Q Denotes The Po-

sition measurement uncertainty at the i-th RP in projection q. This general formulation allows for using any optics model, T, and any number of tracking RPs (greater than 1). A similar approach proved useful already when applied by the TOTEM Collaboration to high β∗optics . Since PPS aims primarily at low β∗optics, further optimizations are possible. Low β∗op- tics is characterized by narrow distributions of the interaction vertices in the transverse plane, σ(x∗) ≈σ(y∗) = O(10 µm). Consequently, the vertex terms in the optics decomposition of Eq. (12) give only a small contribution and can be neglected in the reconstruction without any significant loss of accuracy (cf. Fig. 36, right). This, in turn, can resolve the under-determination of the reconstruction discussed earlier. Since there are only 4 measurements available (2 pro- jections times 2 RPs), only 4 proton parameters out of five (x∗, y∗, θ∗

X, Θ∗

y, ξ) can be determined. Therefore, by default, x∗is fixed to 0, which is a reasonable approximation given the LHC

27

optics used by PPS (low β∗) and the very small x∗RMS in these conditions. In this case, the number of degrees of freedom for the fit is ndf = 4 −4 = 0 and therefore the fit effectively performs a numerical solution of a set of 4 nonlinear equations. It is equally justified to fix also y∗≡0, which results in an alternative fitting model with one less fitted parameter (since ξ is reconstructed from horizontal coordinates) and thus with ndf = 4 −3 = 1. This option has been tried for validation purposes and yields results compatible with those obtained with the default choice.

The general expression in Eq. (22) can be decomposed into a set of simpler equations for the conditions relevant to PPS. The minimum of χ2 from Eq. (22) is described by Eqs. (19) and (20) when the following conditions are met: (i) if two tracking RPs are used per arm (Run 2 con- figuration); (ii) if the proton transport can be approximated by the terms explicitly mentioned in Eq. (12) (a good approximation for 2017 and 2018); (iii) only x∗is assumed to be zero (the case with ndf = 0). Each of these equations gives an explicit expression to determine one of the proton kinematic variables. Only the first equation is nonlinear (ξ on both sides of the equa- tion), whereas the others are linear (ξ is taken from the solution to the first equation). Beyond the usefulness for optics studies as discussed in Section 6, this decomposition can speed up the reconstruction software implementation: there is a single nonlinear equation with a single variable that can be solved in different well established ways, e.g. Newton’s method. Using this optimisation gives results compatible with the full minimisation according to Eq. (22).

During Run 2, PPS was operated with two tracking RPs per arm (denoted “near” and “far”, referring to their position with respect to the IP). The input to Eq. (22) therefore consists of one near and one far RP track, selected such that their combination is consistent with belonging to a proton originating from the IP. The selection is achieved by considering all near-far track combi- nations and retaining only those fulfilling the so called “near-far association” constraints. This selection has a double aim: first, to suppress background, and second, to disentangle multiple forward protons present in the event. The association constraints reflect the expected proton kinematics at the IP (e.g. the RMS of the scattering angles) and the patterns imposed by the LHC optics. For instance, forward protons arrive at the RP detectors at small angles with re- spect to the LHC beam and therefore ∆x and ∆y are expected to be small, of the order of 0.1 mm (∆refers to the near-far difference of the track position). Beyond these, selection criteria based

On ∆Ξ And ∆Θ∗

y are also used, based on the single-RP reconstruction of Eq. (21). The constraints have been tuned using both simulation and data, with the aim of optimizing efficiency and purity. The inefficiency (further discussed in Section 12) can arise either because of overly strict constraints discarding real protons, or overly loose constraints not able to distinguish between two (or multiple) protons in the event. The optimisation of the near-far association constraints is performed for each year. In 2016 and 2017, some of the RPs were equipped with Si strip sen- sors that reconstruct no more than one track per event. In this case, the association constraints can only suppress background and can thus be relatively loose: typically only the ∆ξ criterion with a threshold of about 0.01 is applied. In 2018, all tracking RPs were equipped with Si pixel sensors capable of reconstructing multiple tracks. Disentangling individual protons becomes necessary and tighter constraints are needed: typically ∆ξ (with a threshold of about 0.008), ∆x and ∆y criteria are applied.

The quality of the multi-RP reconstruction can be estimated by propagating the reconstructed protons to the RPs and comparing the positions of the measured and the propagated track impact points; the typical difference is smaller than 1 µm (thus at least an order of magnitude better than the spatial resolution of the RPs).

Figure 20 compares the results of the single-RP reconstruction of ξ from the near and far RPs.

400

Figure 20: Comparison of ξ reconstructed with the single-RP method from the near and far RP in each arm, presented as a function of ξ (fill 5849, 2017). The color code represents per-bin event counts. Left: sector 45, right: sector 56.

2017 (13 Tev)

Figure 21: Mean near-far ξ difference from single-RP reconstruction (in a safe region far from acceptance limitations) as a function of fill number (2017, sector 56). The different colors repre- sent data taken with different values of the crossing angle. The error bars represent the system- atic uncertainty estimated as a difference of means evaluated at two different values of ξmulti.

The difference between the left and right plot follows mostly from the optics difference between sectors 45 and 56. The observed part of the phase space (reflected by the discontinuities in the plots) is limited by the distances of the RPs from the beam at low ξmulti (where “multi” stands for reconstructed with the multi-RP method). The LHC aperture limitations (at high ξmulti, details given in Section 8) and the ∆ξ association cut (e.g. vertical constraints at about ±0.006 in the left plot). Beyond these acceptance limitations, the difference is distributed symmetrically about 0 and is independent of the reconstructed ξ (multi-RP), as expected if the alignment and the optics calibration are correct. An example of the mean difference for multiple fills is presented in Fig. 21. The mean value is stable in time, as expected. The systematic shift between the blue and red markers (different values of crossing angle) can be attributed to a residual miscalibration and represents a measure of the systematic uncertainty of the reconstruction.

Figure 22 shows a comparison of ξ reconstructed with the single-RP and the multi-RP meth- ods. Within resolution, they are expected to give the same results. As expected, the single-RP reconstruction has a rather low resolution. Apart from acceptance limitations (cf. Section 8), the single-multi difference is symmetrically distributed about 0 and has a mean independent

500

Figure 22: Comparison of ξ reconstructed with the single-RP and multi-RP methods, presented as a function of ξ (LHC fill 5849, 2017, single-RP reconstruction from the near RPs). The color code represents per-bin event counts. Left: sector 45, right: sector 56.

2017 (13 Tev)

Figure 23: Mean single-RP vs. multi-RP ξ difference (in a safe region far from acceptance lim- itations) as a function of fill number (2017, sector 56, single-RP reconstruction from the near RP). The different colors represent data taken with different values of the crossing angle. The error bars represent the systematic uncertainty estimated as a difference of means evaluated at two different values of ξmulti.

of ξ, again as expected if the alignment and the optics calibration are correct. A summary of the mean single-multi ξ difference for several fills is shown in Fig. 23. The mean value is stable with time and close to zero (within the estimated uncertainties, Fig. 40). There is a small resid- ual dependence on the crossing angle (colors), which is caused by residual miscalibration and represents a contribution to the systematic uncertainties.

Figure 24 shows an example distribution of the horizontal scattering angle, θ∗

X Distribution Is Expected To Be Symmetric About

zero. Apart from acceptance limitations (cutoffs at the white-blue boundaries) we observe a re- sult compatible with this expectation. Specifically, the mean value of θ∗

X Does Not Depend On Ξ –

a requirement for well calibrated conditions. Figure 25 compares mean θ∗

X From Many Fills. The

mean value is stable over time and close to zero (within approximately ±10 µrad). The small residual dependence on the crossing angle (colors) is again taken as a systematic uncertainty of the reconstruction.

Figure 24: Histogram Of Θ∗

x vs. ξ as reconstructed with the multi-RP method (fill 5849, 2017). The color code represents per-bin event counts. Left: sector 45, right: sector 56.

Figure 25: Mean Value Of Θ∗

x (in a safe region far from acceptance limitations) as a function of fill number (2017, sector 56). The markers in several colors represent data taken with different values of the crossing angle. The error bars represent the systematic uncertainty estimated as a difference of means evaluated at two different values of ξmulti.

Figure 26: Histogram Of Θ∗

y vs. ξ as reconstructed with the multi-RP method (fill 5276, 2016). The color code represents per-bin event counts. Left: sector 45, right: sector 56.

Figure 27: Mean Value Of Θ∗

y (in a safe region far from acceptance limitations) as a function of fill number (2016, sector 45). The markers in different colors represent data taken with different values of the crossing angle. The error bars represent the systematic uncertainty estimated as a difference of means evaluated at two different values of ξmulti.

Figure 26 shows an example distribution of the vertical scattering angle, θ∗

Y Distribution Is Expected To Be Symmetric About

zero. Except the low-ξ region in the left plot (sector 45), which is affected by radiation dam- age (cf. Section 12), we find this symmetry well maintained. A collection of θ∗

Y Mean Values

extracted from several fills is presented in Fig. 27. The mean is stable over time and close to zero (within ±10 µrad). A single value of the crossing angle was used in the pre-TS2 period in 2016, and a different one in post-TS2 one.

The reconstructed proton objects provided for physics analyses combine: • proton kinematics at the IP: deduced from tracking RP measurements (as discussed

Above) And

• proton timing information: determined from timing RPs. The timing information can be used to match PPS protons with a vertex in the central detector and thus for background suppression, cf. Section 13.

Tracks from the tracking and timing RPs are matched using ∆x, the difference between the

2018 (13 Tev)

Figure 28: Association of local tracks from tracking and timing RPs (fill 7039, 2018). ∆x refers to horizontal distance between the tracks from tracking and timing RPs, σ(∆x) stands for the corresponding uncertainty. The vertical red lines delimit the tolerance window.

2018 (13 Tev)

Figure 29: Multiplicity of reconstructed protons per arm and per event (2018 data). The his- tograms are normalised to unit area. Different colors correspond to different fills as indicated in the legend. Left: sector 45, right: sector 56.

x coordinate measured in the timing RP and that interpolated from the tracking RPs, cf. Fig. 28. The shape of the histograms effectively reveals the “shape” of the timing pad, somewhat smeared by the limited resolution of the tracking in the RPs. The tracking and timing tracks are matched if the ratio ∆x/σ(∆x) is between −1.5 and +2.0. This ratio range was determined empirically to provide good efficiency and purity.

Figure 29 shows the multiplicity distributions of protons reconstructed per arm and per event. As expected, the probability decreases with increasing multiplicity. There are almost no events with five or more reconstructed protons.

Figure 30 shows the raw ξ distributions as extracted from data with no selection based on reconstructed-proton observables. Since most of the protons detected in the RPs are due to pileup, they are not related to the triggering event in the central CMS, and the correspond- ing data set has essentially no bias due to the trigger. No corrections (acceptance, efficiency,

2018 (13 Tev)

Figure 30: ξ distributions as extracted from reconstructed protons with no corrections (accep- tance, efficiency, etc.), 2018 data. The histograms are normalised to unit area. Different colors correspond to different fills as indicated in the legend. Left: sector 45, right: sector 56.

unfolding or so) were applied to these distributions. The shape of the distributions is largely influenced by the acceptance, cf. red curves in Fig. 35. The differences between the left and right plots mostly follow from the difference in the optics between the sectors 45 and 56.

Aperture Constraints

Forward protons traveling from the IP to RPs may be intercepted by various LHC aperture limitations (collimators, beam screens, etc.), which result in detection inefficiency. These effects may be studied either by analyzing the aperture constraints of all LHC elements between the IP and the RPs or empirically by searching for discontinuities in the reconstructed distributions of the proton kinematic variables. This section presents a simple study with the latter approach, performed on zero-bias data (no trigger requirement) with limited statistics.

The study is based on the distributions of the reconstructed scattering angles vs. ξ, cf. Fig. 24 and 26. In both projections the data are limited in the low- and high-ξ region. The limitations at low ξ mostly come from the distance of the RP from the beam. This effect can be modelled by considering the distance and the shape of the sensors, as done in Section 9. The limitations at high ξ are especially sharp in the x projection, indicating that the edge arises because of horizontal constraints — a consequence of the large horizontal dispersion. The slope of the

Constraint In The Θ∗

x vs. ξ plane is given by the interplay of the horizontal dispersion and the effective length optical functions at the limiting LHC element. Figure 31 shows a typical high-ξ pattern in the θ∗

X Vs. Ξ Distribution That Features A Disconti-

nuity (green markers), which is qualitatively similar for all fills in Run 2. The discontinuity is extracted by slicing the color-coded 2D histogram at constant θ∗

X And, For Each Slice, Finding

the ξ position of the discontinuity (each green marker corresponds to one slice). In the left plot (sector 45), the results form a two-segment line indicating possibly the presence of two relevant aperture-limiting entities. The red line represents a two-segment line fit:

X = Θ0 + A (Ξ −Ξ0),

a = a0 for ξ < ξ0 or a1 for ξ ≥ξ0.

(23)

In the right plot (sector 56), this simple parametrisation is compatible with the green points within the estimated uncertainty.

Figure 31: Distribution Of Θ∗

x vs. ξ reconstructed with the multi-RP method (fill 6617, 2018), zoomed at high ξ. The color code represents per-bin event counts. The green markers show the identified aperture cutoff, the red line the fit according to Eq. (23). The green error bars vertically represent the bin size, horizontally a combination of statistical and systematic uncer- tainties. Left: sector 45, right: sector 56.

Figure 31 shows a significant asymmetry between sectors 45 and 56. This follows from the asymmetry of the optics; since in sector 45 the horizontal dispersion is larger, the aperture limitation is reached at smaller ξ values.

The fit according to Eq. (23) has been performed independently on data from different fills, different crossing angle and β∗values — in order to assess a possible dependence on these parameters. An example of such a study is shown in Fig. 32. Within uncertainties, we observe almost no fill dependence (time stability) and a linear dependence on the crossing angle, which is expected from the optics dependence, cf. Eq. (17). Equivalent conclusions have been reached for other data-taking periods in Run 2.

Proton Simulation

This section describes a fast simulation of forward protons in PPS. By design, it does not simu- late details (interaction of protons with matter) but focuses on higher-level observables where the reproduction of features of the data is important. In particular, the simulation accounts for

The Following Effects:

• beam smearing at the IP: vertex smearing and angular smearing (i.e. beam diver-

Gence);

• proton propagation from the IP to the center of each RP according to the LHC optics,

Cf. Section 6;

• simulation of the LHC aperture limitations according to the model from Section 8; • proton propagation between sensors in each RP: linear propagation because of the

Lack Of Magnetic Field In The Rp Region;

• sensor efficiencies (optional): using efficiency maps extracted from data, cf. Section

12;

• geometrical acceptance: check if the simulated protons pass through the sensitive

Fill 7315

Figure 32: Summary of aperture-limitation parameters extracted from several LHC fills (dif- ferent colors, 2018) and several crossing angle values (horizontal axis), for sector 45. The error bars represent a combination of statistical and systematic uncertainties.

36

• digitisation: a software “hit” object is created at the nearest strip/pixel — an effective pitch is used to reproduce the spatial RP resolution extracted from data; • for timing sensors, simulation of proton arrival time (with timing resolution ex- tracted from data, cf. Section 13).

The hit objects created in the simulation are then processed with the standard PPS reconstruc- tion software. The simulation can take into account realistic distributions of parameters of importance: β∗, crossing angle, optics, RP positions, apertures, resolution and efficiencies. The values of the crossing angle and β∗are randomly sampled from the 2D histograms extracted from the data, cf. Fig. 2. The variations in RP positions reflect the movements performed during the LHC operation: e.g. vertical RP movements in the technical stops of 2018 to distribute the radiation damage. For consistency between simulation and data, the simulation conditions are randomly switched with the frequency extracted from data (following integrated luminosities).

The simulation can be used with any source of simulated forward protons. By default, the simulation uses a particle gun, which generates protons with a uniform ξ distribution and

X And Θ∗

y distributions with zero mean and RMS of 60 µrad. These settings simulate minimum bias protons. The beam divergence, σbd, used in the simulation was extracted from data using three com- plementary methods. First, the beam divergence can be estimated from the beam emittance,

P

ϵ/β∗. The second estimate is obtained from the beam spot size, σbs measured by the CMS central detector: σbd = σbs

2 Stems From

converting the beam spot size (product of two beams) to the single-beam width, cf. Eq. (10). The third method is the most direct, but can only be applied to data from the special “align- ment” fills where a sample of elastically scattered protons can be selected. In the final state of elastic scattering there are two protons, ideally with exactly opposite directions. Since the direction fluctuations are predominantly caused by the beam divergence, the size of the latter is determined from the RMS of scattering angle differences between the two elastic protons.

All the methods agree on a beam divergence of about 30 µrad. Multiple validations were performed to check whether the simulation reproduces observations; an example is shown in Fig. 33. In the left plot, the simulation describes well the cutoff at low x (because of the sensor edge) and the smooth cutoff at large x (because of the LHC aperture limitations). In the right plot, the simulation describes well the cutoff at large y (because of the sensor edge).

An example of the timing simulation is shown in Fig. 34. Here, a realistic timing resolution is used for the reconstructed protons (vertical axis), but perfect vertex z (horizontal) reconstruc- tion is assumed.

Figure 35 shows the effect of the LHC aperture limitations (discussed in Section 8) on PPS acceptance, which is estimated with the proton simulation. The differences between the left and right plots stem primarily from the differences in the optics in the LHC sectors 45 and 56. The differences between the colors (representing different years) are related to the sensor types used in different years. In 2016, very wide Si strip sensors were used, thus limiting potential loss of protons because of the vertical displacement from the beam. Consequently, the green curve presents a plateau close to full acceptance at the central ξ range. In 2018, vertically narrower Si pixel sensors were used, thus unable to detect protons with sizable vertical displacement from the beam. The proton loss rate increases with ξ due to the optics: in particular due to the

2018 (13 Tev)

Figure 33: Comparison of hit distributions from simulation (red) and LHC data (fill 6738, no explicit event/track selection, black), for the 2018 pre-TS1 configuration and the near RP in sector 56. The black error bars represent statistical uncertainties. Left: distribution of horizontal track positions, right: distribution of vertical track positions (y range limited to the area around the upper sensor edge).

30

Figure 34: Simulated correlation between vertex position along the beam, z∗, and the proton timing difference observed in LHC sectors 56 and 45 (2018 pre-TS1 configuration). The color code represents per-bin event counts. Reconstruction resolution of z∗is not included in this plot. The red dashed line indicates the ideal correlation.

Cms-Totem

Figure 35: Fraction of reconstructed multi-RP protons, as a function of ξmulti, for a proton sam- ple produced with the PPS direct simulation. Since perfect detector efficiency was assumed in this simulation, the results reflect mostly the geometrical acceptance of the apparatus.

nonzero value of Dy (cf. Section 6.2.4) and Ly increasing (in absolute value) with ξ (cf. Fig. 16). In 2017, a hybrid configuration with strip (pixel) sensors in the near (far) RP was used (cf. Table 1) and consequently the acceptance shape is in between the two extremes. The acceptance in sector 56 (right plot) is more sensitive to the reduced size of the pixel sensors because of the larger (absolute) value of Ly in this sector.

Uncertainties

Since the simulation described in Section 9 reproduces the data well (cf. e.g. Fig. 33), it can be used to validate the performance of the proton reconstruction presented in Section 7. The performance will be characterized in terms of the three quantities below.

• “Bias” = mean of reconstruction - truth. This may occur because of effects neglected by the reconstruction formula; a notable example is the single-RP reconstruction of ξ, Eq. (21), which is unable to correct for the effect of the horizontal scattering angle

Θ∗

x. The “bias” may be nonzero even with a perfect knowledge of the conditions (alignment, optics, etc.). • “Resolution” = RMS of reconstruction - truth. This may occur because of random event-to-event fluctuations, e.g. from finite sensor resolution or imperfect separation of kinematics variables in the reconstruction. A notable example of the imperfect separation could be the single-RP reconstruction of ξ; since this reconstruction is

Biased By A Term Proportional To Θ∗

x, the event-to-event fluctuations in the scattering angle effectively lead to a degraded ξ resolution. The “resolution” may be nonzero even with a perfect knowledge of the conditions.

• “Systematics” = effect of biased conditions. The “systematics” may be nonzero even if “bias” and/or “resolution” vanish. The considered sources of conditions bias include: • alignment: following the uncertainties from Table 3, variations of the horizontal and vertical alignment were studied separately. Furthermore, symmetric (same sign in near and far RP) and anti-symmetric (opposite sign in the two RPs) shifts have been studied.

Deployment In Out-Of-Position Situations

D. Bendjaballah1, A. Bouchoucha1, M. L. Sahli1,2* and J-C. Gelin2

Abstract

Side-impact collisions represent the second greatest cause of fatality in motor vehicle accidents. Side-impact airbags have been installed in recent model year vehicle due to its effectiveness in reducing passengers’ injuries and fatality rates. In meeting these requirements, simulations of folding and deploying airbags are very useful and are widely used. The paper presents a simulation method for the deploying airbags using three materials in different working conditions. Finite element analysis is primarily used to evaluate this concept. In these simulations, the gas flow is described by the conservation laws of mass, momentum, and energy. The numerical results indicate that the FE method in this paper is capable of capturing airbag deploying process accurately.

Keywords: Airbag simulations, Out-of-position, Crash, Modeling, Out-of-position

Background

The passive safety of cars has become a very high prior- ity issue for the automotive industry. Today, there are not only one or two airbags in a car; certain models have ten times more than that. With the increasing usage of airbags, the number of accidents where the airbag itself can cause an injury to the occupant also increases

(Augenstein Et Al. 2003; Gabauer And Gabler 2010;

Audrey et al. 2011). As is well known, safety belts are also now devices designed to provide protection to the users of vehicles during crash events, minimizing the loads necessary to adapt their movement to the move- ment of the car (Freesmeier and Butler 1999; Schmitt et al. 1997). In general, the seat belt is designed to restrain the occupant in the vehicle and prevent the

Occupant From Having Harsh Contacts With Interior

surfaces of the vehicles. The airbag acts to cushion any impact with vehicle structure and has positive internal pressure, which can exert distributed restraining forces over the head and face. As a safety component of auto- mobile, an airbag decreases occupants’ injury likelihood effectively in case of an accident (Ruff et al. 2007). These safety elements can reduce the death rates on the roads, and its protection effects have been widely approved (Crandall et al. 2001; Teru and Ishikawa 2003). With computational tools such as finite element methods designed for dynamic contact problems, crashworthiness simulations can now be used with reliable accuracy to evaluate occupant protection in various collision condi- tions with safety metric/parameters such as acceleration, head injury criteria, intrusion distance, intrusion vel- ocity, and neck forces (neck injury risk or whiplash).

Thus, new types of airbag products are being developed to handle different collision scenarios.

Become Standard Equipment On Most New Passenger

vehicles (Braver and Kyrychenko 2004; Teng et al. 2007; Yoganandan et al. 2007). The airbag cushion is com- posed of a woven fabric which is rapidly inflated during a car crash. The airbag dissipates the passenger’s kinetic energy thereby reducing injury through biaxial stretching of the fabric bag and escaping gas through vents. There- fore, the performance of the airbag is greatly influenced by the mechanical properties of the fabric. Generally, air bags are designed to deploy in a crash that is equivalent to a vehicle crashing into a solid wall at 8 to 14 mph.

Air bags most often deploy when a vehicle collides with another vehicle or with a solid object like a tree. There are various types of airbags: frontal, side-impact, and curtain airbags. In general, the passenger side airbags are usually larger than the driver airbags (see Fig. 1).

Besançon, France

© The Author(s). 2017 Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Bendjaballah et al. International Journal of Mechanical

Doi 10.1186/S40712-016-0070-2

Extensive studies have shown that the airbag deploy- ment in load cases consists of two occupant loading phases: a punch-out effect where the airbag bursts out of its container with the airbag and airbag module cover accelerating towards the occupant and a second loading phase during which the airbag is taking on its deployed shape and volume (membrane-loading effect). Bankdak et al. (2002) developed an experimental airbag test system to study airbag-occupant interactions during close proximity deployment. The results provided insight for simulating the effect of inflation energy and mass flow on target response. Bedard et al. (2002) found that while left-side (driver-side) impacts accounted for only 13.5% of all crashes, the fatality rate among these

Crashes Was 68.3% In Comparison To Front Impact

(48.3%), right-side impact (31.3%), and rear impact (38.4%). These studies underscore the importance of oc- cupant safety during side-impact collisions. In the last years, the current market requested to reduce the time and cost airbag development. In order to achieve this result, virtual simulations play an important role since they allow to minimize the number of experimental tests (Pei et al. 2013; Cao et al. 2014). Several simulation models of airbag were established (Wang et al. 2007). It is feasible to optimize the parameters of airbag deploy- ment using simulation technology. Experimental and numerical studies have quantified injury risks to close- proximity occupants from deploying side airbags. These studies have focused on the prevention of the most ad- verse effects of airbag deployment (Duma et al. 2003).

Other studies have proposed airbag characteristics to minimize particular biomechanical responses (Haland and Pipkorn 1996). In a more recent study, Marklund and Nilsson (2003) compared deformation patterns with experimental data as well as the computational costs associated with three different airbag deployment simu- lation methods; they concluded that the SPH method is relatively inexpensive and produces incremental deform- ation patterns that compare most closely to the experi- mental results. The process of inflation of an airbag is one of the determining factors in saving lives. The duration from the initial impact of the crash to the full inflation of an airbag is about 40 ms, and during this time, the airbag goes from being in a folded state to a fully inflated state, with a high internal pressure. After achieving this state, the airbag begins to deflate, thus providing a nice cushion for the body impacting it.

Ideally, the person in the crash should come into contact with the airbag at this time. In the present study, a large volume passenger side airbag model is developed to handle different collision scenarios. The main aim is evaluate the performance of deploying of passenger side airbag using finite element methods (FEM).

Materials

The tensile specimens were made in different airbags (P: Peugeot, R: Renault, and VW: Volkswagen) with a length of 200 mm long and a width of 40 mm. Table 1 shows the mechanical properties of the airbag.

Tensile Tests

To determine the mechanical properties of the material of airbag used in the test pieces, tensile tests were performed on Lloyd EZ20 universal testing machine in Constantine. These tests were conducted using rect- angular samples. The axial force and axial displacement acquired during a test are converted into stress and the strain in order to be used for the fabric material model.

The continuous recording of the stress-strain data was performed during both the load and unload phases. A minimum of five samples were made in order to check the repeatability of the measurements. All the data was collected by using a PC-based data acquisition system and analyzed by commercial software. The picture frame test device that is made for this study is shown in Fig. 2.

Fig. 1 a Frontal and side airbags. b Oblique view of facet occupant model in sitting posture following airbag deployment (Lim et al. 2014)

0.150

Bendjaballah et al. International Journal of Mechanical and Materials Engineering (2017) 12:12

Page 2 Of 9

Figure 3 shows the stress-strain relationship of the airbag sample under axial tensile loads. The results are showing a linear increase in extension with the increas- ing stresses. This is an expected output and it confirms with the theoretical behavior of a sample subjected to tensile stress. The rupture strain values for different airbags (R/P/VW) were 0.322, 0.441, and 0.472, respect- ively. The measured elastic parameters (i.e., Young’s modulus E and initial yield strength) and Poisson’s ratio are summarized in Table 2. The tensile tests of the woven fabrics can show differences on mechanical prop- erties because woven fabrics can resist in-plane shear loads once the yarn lock-up angle has been reached. The differences of material property on material direction can affect the shape of fully deployed bag (see Fig. 3b).

Theoretical Background

Numerical simulations of airbags use very complex and techniques such as an orthotropic model to identify the mechanical behaviors during the airbag inflation and the fluid mechanics (gas flow) to describe the inflator gas flow (pressure gradient) and improve the representation of the pressures within the airbag. To model the airbag as an orthotropic model, three material constants have to be provided. Assuming a plane stress condition, the

Ð1Þ

where σ is the normal stress and τ is the shear stress, the subscript refers to the principal material directions, i.e., the fill and warp directions. Also, ε and γ are the strain components. The material elastic constants Qij are

Ð2Þ

where E1 and E2 are the Young’s modulus in the fill and wrap directions and G12 is the shear modulus of the fabric material. νij is the Poisson ratio of the material.

The gas exerts a pressure load on the airbag causing it to expand. This expansion puts the airbag under tensile stress lowering the expansion rate. In this study, heat conduction and heat transfer is not taken into account.

Fig. 2 A photograph of Lloyd EZ20 universal testing Fig. 3 Stress versus strain using Lloyd EZ20 machine for a three different airbags at 0° and 90° and b VW airbag test specimens at

Different Angles

Table 2 Physical and mechanical properties of the airbag

Page 3 Of 9

In the deployment of an airbag, an inflator supplies high velocity gas into an airbag causing it to expand rapidly. The gas inside the airbag is assumed to be ideal, to be of constant entropy, and to satisfy the equation of state:

Ð3Þ

Here p, ρ, and e are respectively the pressure, density, and specific internal energy, and γ is the ratio of the heat capacities of the gas. The gas flow is described by the conservation laws for mass, momentum, and energy that

Ð4Þ

here, V is a volume, A is the boundary of this volume,

N Is The Normal Vector Along The Surface A, And U

denotes the velocity vector in the volume. Applying Bernoulli’s equation in the case of an ideal gas with

Ð5Þ

Here, the subscript ex denotes quantities at the throat of the tube. Furthermore u, p, and ρ denote the quan- tities inside that part of the tube that is supplying mass.

Materials And Boundary Conditions

The airbag system mainly consists of three parts: the airbag itself, the inflator unit, and the crash sensor or diagnostic unit. Thus, to study the behavior of the airbag using FE simulations, we need to have an FE model of the airbag in the folded position. A FE model of the airbag was used to simulate the test condition as shown in Fig. 5. LS-DYNA® material model FABRIC (MAT_34) is used to simulate the airbag material. It is a variation of the layered orthotropic material model. Additionally, in the LS-DYNA® material model, fabric leakage can be accounted for. However, for this CAB material, the leak- age is almost negligible and therefore no leakage is specified. The mechanical properties can be determined from the physical test. Typical material properties for airbag fabrics are taken as given in Chawla et al. (2004a) (Table 3). These properties are used to simulate inflation process of airbag (see Table 1). The car dashboard is modeled as the rectangular thin plate using a MAT_RI-

Gid Material, And The Degrees Of Freedom Are Con-

strained in all the directions. The similar properties of thermoplastic polymer are assigned for contact purposes. The porosity of the fabric is assumed zero. The nitro- gen gas is taken for inflating the airbag. Properties of nitrogen gas and initial bag conditions are shown in Table 4. The example on which we perform the study is a typical passenger side airbag. The geometric de- tails have been measured from a commercially avail- able airbag. The initial state of the airbag is a closed rectangular whose sides are to be finished to 482 × 635 mm2 and is shown in Fig. 4.

Table 3 Material properties of airbag and rigid plate used in FE

–

Table 4 Initial values used for FE simulation of the swelling of

3.33 × 10−4

Fig. 4 The initial airbag geometry in the form of a rectangular Bendjaballah et al. International Journal of Mechanical and Materials Engineering (2017) 12:12

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